Forms and Means of Generalization in Mathematics

Dörfler, Willibald

doi:10.1007/978-94-017-2195-0_4

Cited by 53 publications

(66 citation statements)

References 3 publications

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“…We think that the process of understanding mathematical concepts would progress along with the symbolizing processes (Dörfler, 1991). In particular, the activities of symbolizing enable children to transit from an operational conception to a structural conception in the sense of Sfard (1991).…”

Section: Didactical Ideas For Overcoming Children's Difficultiesmentioning

confidence: 99%

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

Okazaki

Koyama

2005

Educ Stud Math

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When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of "division with decimals" in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to conceive multiplication and division as a system of operations.

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Section: Didactical Ideas For Overcoming Children's Difficultiesmentioning

confidence: 99%

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

Okazaki

Koyama

2005

Educ Stud Math

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“…Upon reflecting on people's thought processes, we can realize that there is a tendency to generalize. Dorfler (1991) understands generalizing as a social-cognitive process, which leads to something general, and whose product consequently refers to an actual or potential manifold in a certain way. We need to distinguish between two aspects of generalization: seeing the general in the particular, or seeing the particular in the general (Kruetski, 1976).…”

Section: Inductive Reasoningmentioning

confidence: 99%

Monitoring and Guiding Pupils' Problem Solving

Čadež

Kolar

2018

magistra

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“…According to Kaput (2008), the core of algebraic thought contains two essential elements: one is using symbols and problem solving to represent mathematical ideas and the other is generalizing. Here, generalization is defined as a process that can come from or cause special situations (Davydov, 1990;Krutetskii, 1976, Polya, 1957 or as a way of transferring information in order to create a statement that is valid for all objects' properties (Dörfler, 1991). Generalization has also been referred to as a process requiring high-level thinking that improves reflective abstraction (Piaget, 1970).…”

Section: Analytical Frameworkmentioning

confidence: 99%

An Analysis of Middle School Mathematics Textbooks from the Perspective of Fostering Algebraic Thinking through Generalization

2017

EDUC SCI-THEOR PRACT

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The efficient use of generalization in textbooks, which are regarded as a supplementary teaching material and constitute a substantial place in the processes of learning and teaching, is important for effective mathematics learning and teaching, as well as for developing mathematical and algebraic thinking. This research seeks to understand how and to what extent middle school mathematics textbooks include both components of generalization in terms of developing algebraic thinking. This study uses the document analysis approach, a qualitative research method, for collecting data. The tasks and exercises in the textbooks have been analyzed using a coding matrix based on the two basic dimensions of the research' s analytical framework. As a result of the research, the tasks and exercises in the textbooks are seen to sufficiently support the component of Generalizing Arithmetic and Quantitative Reasoning, while partially ignoring the component of Patterns and Functional Relationships/Variables. Moreover, the tasks and exercises that are expected to allow students to explore and express generalizations are seen to have been mostly ignored. In addition, generalizations are observed to have been generally associated with everyday life, in parallel with the curriculum.

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Forms and Means of Generalization in Mathematics

Cited by 53 publications

References 3 publications

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

Characteristics of 5th Graders' Logical Development Through Learning Division with Decimals

Monitoring and Guiding Pupils' Problem Solving

An Analysis of Middle School Mathematics Textbooks from the Perspective of Fostering Algebraic Thinking through Generalization

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